If the value of `m^4+1/m^4=119` then the value of `|m^3-1/m^3|`=

Options

A. 11
B. 18
C. 24
D. 36

Answer

To solve the problem where \( m^4 + \frac{1}{m^4} = 119 \) and we need to find \( |m^3 - \frac{1}{m^3}| \), we can follow these steps:

### Step 1: Use the identity for \( m^4 + \frac{1}{m^4} \)
We know that:
\[ m^4 + \frac{1}{m^4} = \left(m^2 + \frac{1}{m^2}\right)^2 - 2 \]
Let \( x = m^2 + \frac{1}{m^2} \). Then we can rewrite the equation as:
\[ x^2 - 2 = 119 \]

### Step 2: Solve for \( x^2 \)
Adding 2 to both sides gives:
\[ x^2 = 119 + 2 = 121 \]

### Step 3: Solve for \( x \)
Taking the square root of both sides:
\[ x = \sqrt{121} = 11 \quad \text{or} \quad x = -\sqrt{121} = -11 \]
Since \( m^2 + \frac{1}{m^2} \) cannot be negative, we take:
\[ m^2 + \frac{1}{m^2} = 11 \]

### Step 4: Use the identity for \( m^2 - \frac{1}{m^2} \)
We can use the identity:
\[ m^2 - \frac{1}{m^2} = \sqrt{\left(m^2 + \frac{1}{m^2}\right)^2 - 4} = \sqrt{11^2 - 4} = \sqrt{121 - 4} = \sqrt{117} \]

### Step 5: Find \( m^3 - \frac{1}{m^3} \)
Using the identity:
\[ m^3 - \frac{1}{m^3} = \left(m - \frac{1}{m}\right)\left(m^2 + 1 + \frac{1}{m^2}\right) \]
We need to find \( m - \frac{1}{m} \):
\[ m - \frac{1}{m} = \sqrt{\left(m^2 + \frac{1}{m^2}\right) - 2} = \sqrt{11 - 2} = \sqrt{9} = 3 \]

### Step 6: Calculate \( m^3 - \frac{1}{m^3} \)
Now substituting back:
\[ m^3 - \frac{1}{m^3} = \left(3\right)\left(11 + 1\right) = 3 \times 12 = 36 \]

### Step 7: Find the absolute value
Thus, we find:
\[ |m^3 - \frac{1}{m^3}| = |36| = 36 \]

### Final Answer
The value of \( |m^3 - \frac{1}{m^3}| \) is \( 36 \).

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